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Chapter 2: Q. B (page 238)

Use the definition of the derivative and factoring formulas to prove that for any positive integer k, the derivative ofxkis kxk-1

Short Answer

Expert verified

ddxxk=kxk-1

Step by step solution

01

Step 1:Given information

The given expresiion isfx=xk

02

Step 2:Explanation

We鈥檒l need to restrict kto be a positive integer. In this case if we define fx=xk,

we know from the alternate limit form of the definition of the derivative that the derivative f'ais given by

f'a=limx鈫�afx-fax-a=xk-xax-a

Now,

xk-xa=x-axk-1+axk-2+...ak-2x+ak-1

If we put this into the formula for the derivative we see that we can cancel the x-aand then compute the limit.

f'a=limx鈫�ax-axk-1+axk-2+...ak-2x+ak-1x-a

鈬�f'(a)=limx鈫�axk-1+axk-2+...ak-2x+ak-1

鈬�f'a=kak-1

To completely finish this off we simply replace the awith an xto get,

鈬�f'x=kxk-1

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Most popular questions from this chapter

Use the definition of the derivative to prove the following special case of the product rule

ddx(x2f(x))=2xf(x)+x2f'(x)

Prove that if f is any cubic polynomial function f(x)=ax3+bx2+cx+dthen the coefficients of f are completely determined by the values of f(x) and its derivative at x=0 as follows

a=f"'(0)6;b=f"(0)2;c=f'(0);d=f(0)

Use (a) the h鈫�0definition of the derivative and then

(b) the z鈫�cdefinition of the derivative to find f'(c) for each function f and value x=c in Exercises 23鈥�38.

29.f(x)=x12,x=9

Use (a) the h鈫�0definition of the derivative and then

(b) the z鈫�cdefinition of the derivative to find f'(c) for each function f and value x=c in Exercises 23鈥�38.

24.f(x)=x3,x=1

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

f(x)=1x2+1
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